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Lower and upper bounds for the splitting of separatrices of a pendulum under a fast quasiperiodic forcing. (English) Zbl 0867.34035

Summary: Quasiperiodic perturbations with two frequencies \((1/\varepsilon ,\gamma /\varepsilon)\) of a pendulum are considered, where \(\gamma\) is the golden mean number. We study the splitting of the three-dimensional invariant manifolds associated to a two-dimensional invariant torus in a neighbourhood of the saddle point of the pendulum. Provided that some of the Fourier coefficients of the perturbation (the ones associated to Fibonacci numbers) are separated from zero, it is proved that the invariant manifolds split for \(\varepsilon\) small enough. The value of the splitting, that turns out to be \(O\left(\exp \left(-\operatorname{const}/\sqrt{\varepsilon}\right)\right)\), is correctly predicted by the Melnikov function.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37G05 Normal forms for dynamical systems
11J25 Diophantine inequalities
37C55 Periodic and quasi-periodic flows and diffeomorphisms
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References:

[1] G. Benettin, A. Carati, and G. Gallavotti, A rigorous implementation of the Jeans-Landau-Teller approximation for adiabatic invariants, Preprint, August 1995. · Zbl 0907.58057
[2] G. Benettin, On the Landau-Teller approximation for adiabatic invariants, In [Sim97]. · Zbl 0984.70014
[3] L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 1, 144 (English, with English and French summaries). · Zbl 1010.37039
[4] A. Delshams, V. G. Gelfreich, A. Jorba, and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Math. Preprints Series 199, Univ. Barcelona, Barcelona, 1996. · Zbl 0897.34042
[5] A. Delshams, V. G. Gelfreich, A. Jorba, and T. M. Seara, Splitting of separatrices for (fast) quasiperiodic forcing, In [Sim97]. · Zbl 0961.37016
[6] Amadeo Delshams and Teresa M. Seara, An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys. 150 (1992), no. 3, 433 – 463. · Zbl 0765.70016
[7] Giovanni Gallavotti, Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review, Rev. Math. Phys. 6 (1994), no. 3, 343 – 411. · Zbl 0798.58036
[8] V. G. Gelfreich, Separatrices splitting for the rapidly forced pendulum, Seminar on Dynamical Systems (St. Petersburg, 1991) Progr. Nonlinear Differential Equations Appl., vol. 12, Birkhäuser, Basel, 1994, pp. 47 – 67. · Zbl 0792.34053
[9] V. F. Lazutkin, Splitting of separatrices for the Chirikov’s standard map, Preprint VINITI No. 6372-84 (in Russian), 1984.
[10] A. I. Neĭshtadt, The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh. 48 (1984), no. 2, 197 – 204 (Russian); English transl., J. Appl. Math. Mech. 48 (1984), no. 2, 133 – 139 (1985). · Zbl 0571.70022
[11] Carles Simó, Averaging under fast quasiperiodic forcing, Hamiltonian mechanics (Toruń, 1993) NATO Adv. Sci. Inst. Ser. B Phys., vol. 331, Plenum, New York, 1994, pp. 13 – 34.
[12] C. Simó , Hamiltonian systems with three or more degrees of freedom, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., held in S’Agaró, Spain, 19-30 June 1995, Kluwer Acad. Publ., Dordrecht, to appear in 1997.
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